Say a universe is worthy iff it’s worth creating. Let M be the multiverse containing all and only the worthy possible universes. Is M the best multiverse? Here is an argument to the contrary. Somewhere in M, there will be an infinite sequence of universes u1,u2,…, with u2 better than u1, u3 better than u2, and so on, such that these universes differ from one another only in the magnitude of a single minor evil. For concreteness, suppose that in each of these universes, there is a counterpart of me who has a mosquito bite that doesn’t affect his life. In u1, the bite itches for one hour. In u2, it itches for half an hour. In u3, for a quarter of an hour. And that’s the only difference. (Maybe this happens during a time in the person’s life where he does nothing but itch or not itch, and later he forget this time.) Call the universes from this sequence u-universes.

Now, let M* be a multiverse containing all and only the worthy possible universes other than u1.

If M is the best multiverse, M has to be better than M*. But this is far from clear. There is a one-to-one correspondence f between the universes of M and M*, where universes that aren’t in the sequence u1,u2,… correspond to themselves (f(u)=u), while f(u1) = u2, f(u2) = u3, and so on. Then, for any universe u in M, either f(u)=u, or f(u) is just like u, except that my counterpart have a shorter length of itching from the mosquito bite in u than in f(u). How can M be better than M*, then?

Well, M is surely not aggregatively better. How could it be, given the correspondence? We describe the difference between M and M* as follows: M is just like M*, except that if anything it has some more itching.

So if it is better, it must be in virtue of a non-aggregative value. The best proposal, which Klaas Kraay made in response to my arguments in the previous multiverse post, is that M maximally exemplifies the value of diversity. But notice that M and M* exemplify the same diversity of goods. The only difference is that M exemplifies a greater diversity of bads: it contains a u-universe with my counterpart having an hour of itching from the relevant bite, while M* does not contain that u-universe. However, both M and M* contain the same countable infinity of u-universes, and they contain all the same diversity of goods from the u-universes and the non-u-universes.

Haven’t you loaded the dice overmuch by stipulating that the bite “doesn’t affect his life”? After all, most theodicies require goods flowing from evils: empathy-development or character-building or soul-making or what-have-you.

I’m wondering whether any universe at all is worth creating. Of course, this would have a lot to do with defining our terms, but there are scenarios where I think you can argue that a traditional concept of God with certain perfections would not create anything. that’s not exactly a new argument, but consider it along with a particular way to think about “worth.”
So, let’s say that a universe is worth creating iff it makes for a better state of affairs than previously existed. I think if you define it along those lines, then you could press the point about creation being inconsistent with God’s perfection.
I’ve also been kicking around a similar argument that considers whether a morally perfect being would necessarily seek to maintain moral perfection. Again, very similar to an old argument, but I think put in a slightly different light. Can we say that a being is morally perfect if it does not seek to maintain said perfection?
The argument for both cases is very similar. Put in Douglas Adams-ish terms: Would a being with certain properties of perfection want to go mucking about with a universe like ours?

D4M10N:
I take it that your suggestion is that a world with an unjustified evil will not be worthy. I wondered about this. This point is not going to be usable for Donald Turner, because Turner’s view, if memory serves me, is that all worlds better than some cut-off are created. Well, imagine a world that’s really great except for that mosquito bite. That’s going to be about the cut-off, whether or not the evil is justified.
Another move, one not limited to Turner’s version of the multiverse view, is to take on board van Inwagen’s suggestion that sometimes a certain amount of an evil is required for some good, but there is no minimum amount of it. So, suppose that for some good, it is necessary that I itch for some amount of time. But there is no least amount of itching that is needed. That would be enough to generate the infinite sequence I need for the argument. (And the amount of itching doesn’t have to go to zero as we progress in the u-series, like it does in my example.) God is justified in creating a world with an hour of itching, even though half an hour would do just as well, because there is no minimum amount of itching. Then, the worlds will still be worthy (if they’re good in other respects).
Mike:
I think there are pros and cons to creating. There are enough pros that God has good reason to create, and some universes are creatable. But there are cons, too, and hence God has good reason not to create. When there is good reason to do something, and good reason not to do it, that is where a morally good beings makes a choice.

Whether there are pros seems to depend on how we define worth. Let’s say God is perfectly circular, and necessarily so. If we then consider that something is worth creating iff it will increase the circularity of the world, then nothing would meet that description. God is already a perfect circle.
So, likewise, it would seem like God couldn’t make a world better than his own existence makes it. Otherwise, God wouldn’t be the greatest possible being, right? So, if we define a world being worth creating iff it makes for a better state of affairs than previously existed, as far as I can tell we run into this dilemma.

Hi Alex,
Thanks for another stimulating post on this topic.
I’ll start with a small clarification. In the earlier thread, I didn’t suggest that a universe like M would maximally exemplify *diversity*. I tentatively proposed, instead, that M is *complete* in a way that (in your example) M* is not.
M* fails to include a universe, U1, which has been stipulated to be worthy of being created and sustained by God. So M* is incomplete: it fails to include *all* worthy universes.
For the record, I’m not yet convinced that no plausible case can be made for the claim that M exceeds M* in aggregative value.
But suppose I’m wrong about this.
What do you think about comparing, not the goodness of the *outcome* of the relevant world-actualizing actions, but instead comparing the goodness of the *actions* of two rival world-actualizers?
World-actualizer A picks M*, and so fails to create a universe, U1, which is worthy of being created and sustained. World-actualizer B picks M, and so does not fail to create any such universe.
It seems that we have good reason for thinking that B’s *action* is better than A’s, even if we grant that the *outcomes* of their actions do not differ in aggregative value. For this reason, the action we should expect an essentially unsurpassable being to perform (assuming multiple universes are possible) is to actualize M.

Klaas:
1. Sorry about diversity/completeness.
2. That said, in M* we do have completeness of types of goods–every type of good that can be instantiated in a worthy universe is instantiated. What we don’t have in M* is completeness of types of worthy universes.
3. The suggestion that one measure the value of actions rather than the value of outcomes is an intriguing one here. I was only addressing the values of the universes in my post.
4. I continue to be worried about numerical identity issues. I assume that on your view, in the multiverse, I exist in only one universe. But there is a possible universe where I am a biologist instead of being a philosopher. And that universe is not in fact in the multiverse. This suggests to me that you only get to say that M contains every type of possible worthy universe, not every possible worthy universe. (Of course, you can embrace counterpart theory, but that too has its costs.) And I find it less compelling that B’s action is better than A’s simply because there is a type of worthy universe that A did not actualize. I could also say that A’s action is better than B’s because there is a type of itch that A did not actualize. So, in one respect, B’s action is better than A’s, and in another, A’s is better than B’s.

If by supposition u1 is worth creating, isn’t its aggregate value positive? Are there universes with negative (or zero) aggregate value that aren’t worth creating?
If u1 has positive aggregate value, a case could be made that M > M*. Consider the following two sums:
S1 = 1+2+3+…
S2 = 2+3+4+…
Now, by your argument, since the nth value of S2 is greater than the nth value of S1, S2 > S1 (or at least S2 >= S1)
But S1-S2 = 1, which is greater than zero, thus S1 > S2.
[The mathematical problem is how to give a reasonable number to a divergent sum.]
Could you say that M must be greater than M* because M* omits a universe with a particular positive goods-to-bads ratio?

A quick point that echoes a post by Mike Almeida in another thread and relates to Klaas Kraayâs âFor the record, I’m not yet convinced that no plausible case can be made for the claim that M exceeds M* in aggregative value.â
The original argument of this thread is assuming a Cantorian notion of size. But Euclidean notions of size (the part is smaller than the whole) are used all the time. Here is an example where M is better than M* (though the numbers donât quite work for the itch example, and imagine that M is just the uâs): Let u1, u2, u3, u4â¦ have values 1/2, 1/4, 1/8, 1/16â¦ Then M is valued at 1, M* at 1/2, and M is better than M*. The bijection from u1 to u2, u2 to u3,â¦ is simply irrelevant. M is better than M*, on the Euclidean notion of size, simply put, because M has more good things.
For those interested in reading more about Cantorian versus Euclidean size, two very good places to start are: 1) Mancosu, P. Measuring the Size of Infinite Collections of Natural Numbers: Was Cantorâs Theory of Infinite number Inevitable? The Review of Symbolic Logic, 2, 2009, 612-646; and 2) Parker, M. Philosophical Method and Galileoâs Paradox of Infinity. Where I believe a copy is still at:http://philsci-archive.pitt.edu/archive/00004276/

CliveStaples:
I don’t see why S1-S2 = 1. Sure, you can write S2 = 1 + S1. But you can also write S1 = S2 + (1 + 1 + 1 + …). With infinities, that’s how it is. 🙂
Jeremy:
“though the numbers donât quite work for the itch example”
Exactly. 🙂
Now consider your case.
M’s value = 1/2 + 1/4 + 1/8 + 1/16 + …
M*’s value = 1/4 + 1/8 + 1/16 + 1/32 + …
Sure there are many bijections f from the worlds of M to the worlds of M*. But there is no bijection f satisfying the inequality: Value(f(x)) >= Value(x), like there is in my example.
Using the standard conventions for handling infinities, we have:
Theorem: If s(1),s(2),… and s*(1),s*(2),… are two sequences of non-negative real numbers, and there is a bijection f between the two positive integers and the positive integers such that s*(f(i)) >= s(i) for all positive integers i, then s(1)+s(2)+… <= s*(1)+s(2)+….
The reason is that commutativity holds for infinite sums of non-negative summands.
Now let’s consider the “part is smaller than the whole” principle. Imagine three infinite sequences of worthy universes that all differ (within the sequences and between the sequences) in ways that are insignificant with respect to value and only in such ways, and that have no individuals in common. The sequences are:
u1,u2,u3,…
v1,v2,v3,…
w1,w2,w4,…
Let:
M be the multiverse u1,u2,u3,…
M* be the multiverse u2,u3,u4,…
N be the multiverse v1,v2,v3,…
P be the multiverse w1,w2,w3,…
Let V indicate aggregative value. Then, plausibly:
1. V(M) = V(N).
After all, if the differences between the components of M and N are value-insignificant, the aggregative differences between the wholes should likewise be. Similarly:
2. V(M*) = V(P).
Again, the differences between the components are value-insignificant. And for the same reason:
3. V(N) = V(P).
And then, by transitivity of value-equality, we get V(M)=V(M*).
So, if you take the Euclidean way, you need to deny 1, 2 or 3. But the rationale for each is exactly the same: in each case we have two infinite sequences of universes, with no universes in common, each of the same individual value, and it’s plausible that the values are the same.
So, either 1, 2 and 3 are all true, or they are all false.
Therefore, to deny V(M)=V(M*) and save the Euclidean principle, we should say that they are all false.
So what holds instead? Do we have V(M) < V(N) or V(N) < V(M)? Since the differences between components are insignificant, and there is nothing in common, we cannot choose one or the other–we have to say: neither.
I think the only way to defend the Euclidean principle is if one thinks that two universes with value-insignificant differences (eyecolor, maybe) and no individuals in common are always incommensurable. I actually think this–I think a world just like this one, but where someone else lives the life of Socrates (if that were possible) would have a value incommensurable with the value of this world. This leads to widespread incommensurability. Few people accept such a consequence. But I like it.
Given such widespread incommensurability, one can uphold the Euclidean principle and say that M is better than M*, as long as individuals are universe-bound (no individual exists in more than universe). In such a case, one can deny that the universe in M is worse than the corresponding universe in M*, because the two universes (e.g., u2 and u3) have different individuals and hence are incommensurable.
But now here is a puzzle. If different universes have no individuals in common, why should different multiverses have individuals in common? And if they don’t, then radical incommensurability comes back to bite one, since M and M* will be incommensurable.

I think the best world is the one that God wants to actualize. In Christian theism we are used to thinking that God is love, but itâs not less true that (as the Qurâan says) God is beauty. In an artistâs creative work itâs not so much that the artist visualizes several end-results, assesses their relative value, and then chooses to realize the most valuable one or the most beautiful one. Rather, it is the power of the artistâs creative will which instills worth and beauty in the final object. So, similarly, the worth and beauty of the actual world are not independent, indeed have no intrinsic existence, beyond Godâs continuous creative will.
Having said that, it is still the case that one way *we* may reason about the kind of world that God would want to actualize is by using our sense of perfection. I think we can rely on our sense of perfection because it is plausible to believe that God, who wants us to be able to know Him/Her, has created us with such a sense. Now in this context I have the clear impression that perfection is more about quality and less about quantity. So itÂ´s not like a multiverse of a billion universes is greater or more perfect or more beautiful or more valuable or more worth creating than one particular universe out of that lot. Nor is it like that if God wants to love a billion created persons then God will want even more to love a trillion created persons. Nor do I envisage the greatest conceivable being having an accountantâs mindset and counting or budgeting mosquito bites, or indeed a micromanagerâs obsession with controlling the tiniest details. Actually, to go back to the artist analogy, I think it is the case that an artist uses randomness in her work, bends randomness to her will and makes it productive. The artist enjoys surprises. Similarly, I am inclined to think, God would want not only to actualize a world with significantly free persons, but also with a âfreeâ (in the sense of non-deterministic or partially unpredictable) impersonal nature. Where perhaps the analogy breaks down is that whereas a human artist assumes risks, God doesnât, in the sense that it is metaphysically impossible for Godâs desires to ultimately fail to materialize in the created world.

In response to Alexâs post of July 1st (12:31am):
Concerning point 2: It may be possible to resist your claim that M* includes *every* type of good that can be instantiated in a universe. Universe u1, which is present in M but absent from M*, must be large and complex enough to feature whatever is required for it to include (a) a mosquito, and (b) a Pruss-counterpart who experiences that mosquitoâs bite for one hour. You say that the *only* difference between u1 and the other universes in the u-sequence is the duration of the relevant itch. But this canât be quite right: there must be a causal story unique to each universe in the u-sequence which explains â perhaps determines â the duration of the relevant itchiness in each universe. (This may involve the physiology of the Pruss-counterpart, or of the mosquito, or, or â¦) Now, perhaps, for all we know, something causally âupstreamâ from the itch in u1 is, or requires, or entails, a good-type instantiated in no other universe. For that matter, perhaps something causally âdownstreamâ from the itch in u1 is, or requires, or entails, a good-type instantiated in no other universe. Skeptical-theism-type considerations might be appealed to here. Given that u1 has been stipulated to be worthy of being created and sustained, it seems epistemically possible that u1 features some good-type not found elsewhere. At the very least, Iâm not sure that youâre entitled to simply stipulate that u1 does not feature some unique good-type: some argument is required.
Concerning point 4: Yes, I incline towards counterpart theory here. Youâre of course right that there are costs, but my sense is that they are worth paying. (The whole multiverse model of theism evidently has its costs, but it seems to me that the benefits may outweigh them.) You said: âAnd I find it less compelling that B’s action is better than A’s simply because there is a type of worthy universe that A did not actualize. I could also say that A’s action is better than B’s because there is a type of itch that A did not actualize. So, in one respect, B’s action is better than A’s, and in another, A’s is better than B’s.â Suppose I grant that there is one respect in which Aâs action is better than Bâs: surely this is a far less significant respect than the respect in which Bâs action surpasses Aâs. B, after all, creates an entire universe (which is worthy of being created and sustained) that A does not.
Finally, it occurs to me that your scenario may need to be tweaked, since one might plausibly deny that your sequence of u-universes is possible. Your scenario posits that there is no smallest possible time period during which an itch can be experienced by a human creature. But this strikes me as quite implausible. After a certain â perhaps vague â point, I think, Pruss-counterparts are just not going to be able to experience sensations of such extremely short duration.

Mike,it would seem like God couldn’t make a world better than his own existence makes it
This is something I’ve thought about, and it doesn’t strike me as correct. It seems to me that x could be a greatest possible being even if a situation in which x exists isn’t a greatest possible situation. It seems to me that a situation containing a greatest possible beings plus lesser beings engaged in a long-term story of relationship and redemption is better in important respects than one that contains nothing but a greatest possible being.
(Incidentally, I’m skeptical that there could be a situation that’s the greatest possible–just add another person to it…)

Klaas:
Good points, as always!
Re. 2: Well, we can imagine that the cessation of the itching is quantum-random, like radioactive decay. Or perhaps God miraculously ends the itching, at a different time in each u-universe.
Re. 4: “B, after all, creates an entire universe (which is worthy of being created and sustained) that A does not.” Note, though, that this universe may have pretty low value–it might be just barely above the cut-off line, with or without the itch.
As for counterpart theory, I wonder if that fits with the kinds of comparisons you want to make. Suppose God creates M. Let x1, x2, x3, … be the relevant itchy person in u1, u2, u3, …, suffering the itch for a1, a2, a3, … units of time, respectively. Now imagine the following plausible counterfactual:
(*) The multiverse would have been better, or at least no worse, had everything been as before, but had x1 suffered a2 time units of itching, and had x2 suffered a3 time units of itching, and had x3 suffered a4 units of itching, and …
(Imagine x1, x2, … collectively coming to complain to God: why didn’t you make him (x1) have u2 time-units of itch, and him (x2) have u3, and so on?)
But the antecedent of (*) describes M*. For the itchy fellow in M*’s u2 can be taken to be the counterpart of the itchy fellow in M’s u1, and so on.
Re. final remark.
Good point. I think we can imagine a being that doesn’t have such limitations. Or we can suppose this: the itch in u1 is an hour long. In u2 it is 45 minutes long. In u3 it is 37.5 minutes long. And so on. (The sequence converges to 30 minutes.)

Joshua,
Iâm also skeptical of greatest possible situations for similar reasons. However, I think what Iâm suggesting could be the answer to that worry. If only God existing is the greatest possible situation, then thereâs only one way for that to happen.
âThis is something I’ve thought about, and it doesn’t strike me as correct. It seems to me that x could be a greatest possible being even if a situation in which x exists isn’t a greatest possible situation. It seems to me that a situation containing a greatest possible beings plus lesser beings engaged in a long-term story of relationship and redemption is better in important respects than one that contains nothing but a greatest possible being.â
Iâm not sure my position is ultimately defensible, but Iâm testing the waters. A couple of thoughts strike me here. If we accept that the lesser beings were created by the greater being, then is there some culpability? We can say âoh, itâs just not the greatest situation, but still the greatest beingâ but itâs the being who brought that situation about in the first place.
I would also say this â compare two beings in possible universes. One being (x), in order to bring about a better state, creates lesser beings as youâve described. The other being (y) decides not to create anything else because the universe is already as good as it can be. Thatâs how I see our positions. Iâm saying that, if itâs true that the situation could be made better (and, thus, is âworthâ creating), then the being wasnât as great as possible in the first place.
What do you think? Hopefully Iâve restated that a little differently to draw something out that maybe didnât come through initially.

Alex,
One way to model the Euclidean notion of size is with hyperreals. Let a sequence of worlds, W, have values w1, w2, w3â¦. Then define the aggregate value of the world, V(W) as the hyperreal comprised of the sequence the partial sums, s1, s2, s3â¦ So if the sequence of values for worlds w1, w2, w3 is 1, 2, 3,,â¦ (that is, the worlds get better), then the value of W = [(1, 3, 6,â¦)]. Removing worlds is equivalent to replacing them with 0âs in the original sequence. So the value of W with w1 hacked off means we have [(0, 2, 5,â¦)], which is smaller/worse. (I think this will work here. Haven’t really thought it through, but I do something similar in a forthcoming paper.)
To answer your question above, V(M) > V(M*). And likely V(M) = V(N) = V(P). Note that on the Euclidean view relabeling is not permitted. If I recall, Mancosu has a good discussion as to why GÃ¶delâs argument (that relabeling is unproblematic) fails.
Also, I wasn’t exactly sure what you mean by value insignificant, so the above may need some tweaking.

Concerning Alex’s point about counterparts (from July 3rd):
I may be missing something here, but I don’t quite see what the problem is. I assume that it isn’t that the itchy people’s complaint to God is legitimate – surely it’s not. I take it that the objection, instead, is that there’s no sensible way to make good the claim that (say) X1’s itch could have had a shorter duration.
But, on the version of counterpart theory I’m exploring in this context, to say that X1’s itch could have been shorter *just is* to say that there is another universe within the multiverse in which X1’s counterpart (X2) has an itch of shorter duration.
On the view I’m exploring, there are no genuinely possible worlds other than the theistic multiverse – the multiverse which God furnishes with all and only the universes worthy of being created and sustained. When we say things like “God might have made fewer worthy universes, or more unworthy universes” we’re – at best – speaking of epistemic possibilities, until of course we see what what we’re describing is not genuinely possible.

A quick comment on the first point in Alex’s July 3rd post:
The scenarios you posit for the cessation of the itching (quantum decay and God’s miraculous intervention) may suffice to guarantee that nothing causally *upstream* from the itch is (or requires, or entails …) a good-type instantiated in no other universe. But they’re not sufficient to guarantee that nothing causally *downstream* from the itch is (or requires, or entails …) a good-type instantiated in no other universe. So I still think that more argument is needed to show that U1 does not feature some unique good-type.
(I should add, though, that nothing much turns on this for my purposes. U1 has been stipulated to be worthy of being created and sustained, and nothing in that description, so far as I can tell, requires there to be a completely unique good-type in U1.)

Klaas:
There may be nothing downstream from the itching. All the causal consequences may simply be removed miraculously.
I wonder if you can make sense of this counterfactual, which I think you need: “If God made M* instead of M, there is a worthy universe that he would have omitted from making.” What makes that true? If the multiverse defines the possibilities, then if God made M* instead of M, then u1 wouldn’t have been possible.

Jeremy:
1. Yes, you can do this, but you need a non-arbitrary ordering of the worlds w1, w2, … (rather, than, say, w7, w3, w200, …) for the partial sums not to be arbitrary. But worlds do not in general seem to come in any single natural order, though in some special cases (like my itch case) they do.
Consider, for instance, worlds that differ in respect of the exact RGB colors of the two eyes of a certain cat whose eyes are never observed by anyone. Any world in that set can be identified by a six-tuple of real numbers between 0 and 1, describing the RGB color of the left eye and the RGB color of the right eye. But in what order do we do the partial sums when we count up the total value of these worlds? The Axiom of Choice (but is it true?) guarantees that there is a total ordering on the set of six-tuples of reals from [0,1], but that total ordering is far from unique.
2. Maybe you can say that M2 is absolutely better than M1 iff M2 is better than M1 regardless of which ordering is chosen for the partial sums. That seems like a really good move. In general being absolutely better than will then be a partial ordering. I think being absolutely better than will not match people’s intuitions, though, unless they have the kinds of incommensurability views that I indicated. (Imagine two infinite sequences of worthy worlds, u1,u2,… and v1,v2,… and suppose that V(vi)=2V(ui) for all i. It will not be the case that the multiverse composed of v1,v2,… is absolutely better than that composed of u1,u2,….)
3. Anyway, a problem with all this heavy-duty set-theoretic machinery is that there is no set of all possible worlds if actual infinities are possible (argument: for any cardinality K, there will surely be a world with exactly K angels).
4. There will also be uniqueness issues in the hyperreal construction itself, though apparently given the continuum hypothesis there is a canonical way of doing it. (But do we have reason to think the continuum hypothesis is true?)
5. “And likely V(M) = V(N) = V(P).” Why?

1. For now, letâs stick to the countable case. (Work is being done extending Euclidean size to the reals, but I havenât worked through it yet. Also, given 3, the reals may not even matter anyway.) Wonât the partial sums, on any rearrangement (that includes all of the wiâs = w1, w2, w3â¦), only differ on a finite initial segment? And, if the difference is only on a finite initial segment, then reordering is no problem. The aggregate values are equal on rearrangements that are equal at a cofinite number of places in the hyperreal.
2. See 1.
3. I donât entirely follow, but I have wondered: is the idea that all of the worthy worlds are a class and not a set? (I agree though, that this worry may very well blow it all out of the water in this context. But itâs still not clear to me that your Cantorian reasoning is any âbetterâ. Simply put, it is plausible that taking a worthy world away from any collection of worlds, set or class, may make the overall worse. I think this plausible whether it can be modeled or not.)
4. Big uniqueness problems. Consider [(1, 3, 5, 7, 9, 11â¦)] versus [(0, 4, 4, 8, 8, 12,â¦.)]. Then the nonprinciple ultrafilter âdecidesâ which is larger. Trying to deal with this issue is where a lot of the current scholarship is. (Itâs all good fun, since really, Cantor is just so 20th century. Euclid is old school.)
5. Assuming that M, N, and P are indexed by the full set of positive integers 1, 2, 3,â¦, and that the individual worlds donât differ in value, then they are the same.

Your argument depends critically on reality being infinitely divisible, and therefore the existence of an infinite number of worthy universes.
1. If a unit of time is not infinitely divisible, then you couldn’t construct your infinite sequence of mosquito bites.
2. If reality in general is not infinitely divisible, then for any bounded size of individual universes, the set of all possible universes at or below this size is finite.
3. Even if time were infinitely divisible, human minds have only a finite number of states. There would reach a point at which you could not distinguish between shorter and shorter periods of itching. That is, there is an N s.t. (assuming that the bad derives from the sensation of itching) you can’t distinguish between the u_n’s for n > N.
It seems to me that the issue you’re discussing is related to the many weird results you get when you start mucking about with infinities. It is precisely these issues that leads many thinkers to conclude that the universe is not infinitely divisible.

Alex,
Youâre quite right that there are going to be tricky issues here. The simplest reply for me to give, concerning your counterfactual, is that it should be taken to have a necessarily false antecedent: âIf, per impossibile, God had failed to create all the good universes â¦â
But the worry goes deeper, and could be expressed as follows. If Iâm trying to ground all modal claims in universes instead of worlds, then this conditional should be read as: âIf, per impossibile, there were a universe in which God fails to create all the good universes â¦â. Of course thereâs no such universe, but thatâs not very interesting.
It occurs to me that there might be a structurally similar problem for Lewisâ modal realism. We might want to say, for instance, that things could have turned out differently than Lewis thought: for example, there might have existed one less concrete possible world than his model allows. But itâs not clear how to express that thought within the confines of his system.
Iâm not sure how best to handle this worry. Certainly it would be nice to avoid having to say that there is one way of understanding modal claims for universe-bound individuals, but another way for God.

Jeremy:
4) How can you talk about ‘the’ non-principal ultrafilter on either of those infinite sets?
Also, how do you compare the sequences of partial sums? Typically in analysis the limit of such a sequence is used, but few cases here involve convergent sequences of partial sums.
I suspect that there are results in measure theory that are more applicable than trying to cram things into (what seem to me to be) Riemann sums. Lebesque, I think, is probably more useful than Euclid here.

Jon:
Well, one might imagine a infinity of logically possible beings who are sensitive to smaller and smaller lengths of itching. It seems that for any t>0, it is possible for there to be a being that can notice t seconds of itching. (Maybe, though, you could worry that the itches are worse for the more sensitive beings? I suppose one could stipulate that away.)
Alternately, instead of shortening itches, one can lengthen the duration of a minor pleasure had by Bambi. In u1, Bambi enjoys some pleasure for an hour, in u2 for two hours, in u3 for three hours, etc.

Alexander,
My point is simply that your argument assumes that the set of worthy universes is infinite. Otherwise you cannot construct the ordered infinite subsequence used to construct a strictly better multiverse.

Isn’t it pretty plausible that there are infinitely many worthy universes?
Consider a worthy universe with just n happy angels. Then a universe much like it but with an extra happy angel will surely be worthy.

CliveStaples:
By ‘the’ I meant whichever ultrafilter happens to be in use. Perhaps that was misleading and I should have written ‘a’. Of course there are very many. (2 to the continuum? — mind’s a bit slow today.)
‘Also, how do you compare the sequences of partial sums?’
As Alex mentioned, the ordering comes from the ultrafilter. You see which holds ‘almost everywhere’: less than, greater than, or equal.
However, as Alex pointed out, rearrangement is providing some hurdles for good old Euclid here in this context.

Alexander,
It’s unclear to me that if a universe with n happy angels is worthy, then one with n+1 happy angels is also worthy. The second universe is larger, and maybe there is a size limit on worthiness.
A possibility that seems plausible to me is that there is only one worthy “seed” universe, and the set of worthy universes is the multiverse generated from that initial starting point.
P.s. Are angels inside the “universe” anyway?

Jeremy/Alexander:
Selection of the ultrafilter seems arbitrary. Is there a ‘canonical’ ultrafilter, or some reasonable method of selecting one? Or is this a fruitless path?
Alexander/Jon:
Perhaps an argument could be made along a continuum. Suppose that the happiness of a given random angel is given by a probability density function f(x). Then there are uncountably many universes where the probability density is shift slightly toward the maximum amount of happiness (or the least upper bound, if no such maximum exists). This would require no additional size or increase in the number of angels.
Or perhaps a “minimal criminal” argument would work. Let h denote the happiness of the least happy individual in a given worthy universe U. Let U* denote a universe identical to U, but where h* (corresponding to h) is in the open interval (h, h+r) for some r>0. U* is also worthy, and entails no greater size. Thus there are at least 2^(aleph-0) worthy universes.

Jeremy:
“Simply put, it is plausible that taking a worthy world away from any collection of worlds, set or class, may make the overall worse. I think this plausible whether it can be modeled or not.”
I feel the force of that principle, but this principle only works if one can numerically identify entities between multiverses, so one can say things like: “u1 in M is numerically the same as u1* in M*.”
If all one has is qualitative identity–exact likeness–then the principle may not be right. Consider a multiverse M which contains aleph0 duplicates, U1,U2,…, of a universe U, in each of which the one and only person named “Smith” itches for an hour (the reason for this definite description is because we aren’t having identity across worlds). It is very plausible that it is true at M that “Things would be worse if the person named ‘Smith’ in U1 were to itch for two hours.” Let M* be the multiverse where this happens. This is a multiverse that contains aleph0 duplications U1*,U2*,… of U, plus a new universe-type, V, which is much like U, but where the person named “Smith” itches for two hours. So, M* is supposed to be worse than M. However, if you can’t numerically identify universes between multiverses, then:
M* is simply a multiverse with aleph0 copies of U and one copy of V
M is simply a multiverse with aleph0 copies of U.
And it now looks like M is M* minus a universe, and if V is worthy, which it could well be, we have a counterexample to the principle: M* is worse than M, but by the principle you give it’s better.
This argument does require the possibility of multiple qualitatively identical universes in one multiverse. Klaas wanted to leave that possibility open, I think. Anyway, I can run a more complicated variant of the argument without assuming this.
So, if there is no numerical identity between different multiverses, the principle doesn’t work.
But if there is numerical identity between different multiverses, then it’s extremely plausible that there is more than possible worthy universe that contains me, and so the multiverse that contains all possible worthy universes contains me in a number of universes, which is paradoxical and makes my life choices meaningless. If in this worthy universe I exercise, in another one I slack off. So why bother exercising?

I’m not clear on why you’re considering duplicates of universes. Why should any (worthy) multiverse contain copies of any universes? If it’s better to have k copies (where k is a cardinal number, possibly transfinite) of a given worthy universe U, wouldn’t it be even better to have 2^k copies?
Is there truly a worthiness/value difference between the multiverse comprised only of the single worthy universe U, and the multiverse comprised of k [i]exact[/i] copies of U?

I suppose it’s really a question about identity, as you said above. Is there really ‘additional’ value, or is it just the same value? Suppose that John doing act A in universe U increases the value of U. If there is another copy of U, then it’s the same John doing the same act–like watching a video repeat of the act. The essential value that U would add to a multiverse it is contained in is fully expressed in the original version.
That doesn’t really amount to providing good reasons to suppose that copies don’t increase aggregate value, so I guess I’m not being terribly philosophically nuanced, but then my training is in mathematics. C’est la vie.

Alex:
In your Smith itching example, to âget fromâ M to M*, do you change u1, or do you add another universe, V? As far as I can tell you are switching between the two, which is fine/equivalent if youâre Cantorian. On the Euclidean view, these are not the same. The number of numbers 1, 2, 3, â¦ is one more than the number of numbers 2, 3, 4â¦ and one less than the number of numbers 0, 1, 2, 3â¦. (Or am I misreading your example?)
CliveStaples:
There is no canonical ultrafilter, there is no reasonable method of selecting one, and this is not, in general, a fruitless path (or so I suggest). I reread my paper related to this topic and what I do is the following. Imagine that you get $1 a day forever. Versus $2 a day forever. Let the value of these offers be the hyperreal of the partial sums. Then we can compare offers. [(1, 2, 3â¦)] is less than [(2, 4, 6â¦)]. The second is better. Then, imagine that you have the offer of 2 dollars a day or 4 dollars every other day, where with initial values, the way it works out is the comparison between [(2, 4, 6, 8, 10â¦)] and [(1, 5, 5, 9,â¦)]. Then the (an) ultrafilter âdecidesâ which is better. And I suggest that this is exactly what we want. Itâs really not clear which is better. The indeterminacy in the ultrafilter mirrors the intuition that each might be better than the other. (One problem is, what if they are equally good? But note that they are infinite values, and they only differ by 1 one way or the other. So the ultrafilter, I think, is doing pretty well.)
The trouble with comparing this procedure with the multiverse context is that the money example clearly deals in the realm of the potential infinite, whereas the multiverse is an example of the actual infinite. With the money, time and potentiality gives a natural ordering, whereas with a multiverse, rearrangement is causing trouble thus far. In case there is any interest, a copy of the paper is here (where the money example begins on the bottom of page 5):http://gc-cuny.academia.edu/JeremyGwiazda/Papers/646119/

Jeremy:
“On the Euclidean view, these are not the same.”
Do you mean that there are TWO different multiverses, maybe M* and M**, one of them obtained by taking M and removing u1, and the other obtained by taking M and upgrading u1 to u2, u2 to u3, and so on?
If so, how do M* and M** differ?

Jeremy:
One more thought. I really do think, as you suggest, that your money example is only tenable insofar as there is a background, a “receptacle” (in the sense of the Timaeus), for the gifts of money–namely, time.
Consider three scenarios:
S1: The universe comes into existence on day 1, and you get one unit of happiness on day 1, two units of happiness on day 2, three on day 3, and so on. You’re the only person in existence, and the doses of happiness are all that happens to you.
S2: Just like S1, except that each day’s dose of happiness has one more unit in it.
S3: There is no day 1. The universe comes into existence on day 2, where you get two units of happiness, then on day 3, you get three units of happiness, and so on.
Now, if we take an absolutist, “Newtonian” view of time, then we can distinguish S2 from S3 in terms of happiness: after all S2 contains an extra day of happiness, namely day 1. But if we take a relationalist, “Leibnizian” view of time, there is no absolute “day 1”–there are only the relations between the days. On such a view, S2 and S3 are exactly the same–there is no way of distinguishing S2’s “day 1” from S3’s “day 2”.
But the universes in a multiverse have no receptacle–there are no spatiotemporal relations between universes. They are thus more like Leibnizian days than like Newtonian days.

Alex:
Re 1st post: At the least, I am suggesting that these two are different. In both, begin with M = u1, u2, u3â¦ In M* add V. In M** change u1 into V. M* has one more universe. Just like 1, 2, 3â¦ has one more integer than 2, 3, 4â¦
The two cases you describe are different if the upgrades occur for all of the uiâs. The difference would be that M* would contain 1 less ui than M**. But if the upgrades don’t all ‘go through’, then they are the same. M* and M** would each lose a universe relative to M.
Re 2nd: Good points. Iâm inclined to agree all round, though also working on some things.

Jeremy:
I just don’t see how there can be a difference between M* and M**.
Consider the following plausible principle:
D. Suppose that A and B are two objects, and the Xs and Ys are pluralities, such that:
D1. A is a mereological sum of the Xs.
D2. B is a mereological sum of the Ys.
D3. No two of the Xs are duplicates.
D4. No two of the Ys are duplicates.
D5. Each of the Xs is a duplicate of exactly one of the Ys.
D6. Each of the Ys is a duplicate of exactly one of the Xs.
D7. A is not a duplicate of B.
Then: The Xs are arranged differently in A from how the Ys are arranged in B.
If I have two Lego constructions, and each construction is made from a bunch of pieces no two of which are alike, and yet each piece in one construction is just like a piece in the other construction, and if the two constructions are not exactly alike, then the pieces must be differently arranged in the two.
Now, suppose that A=M*, B=M**, the Xs are the universes of M*, and the Ys are the universes of M**. Then D1-D6 are verified. If, as you claim, M* is not a duplicate of M**, it follows by D that the universes in M* are differently arranged from those in M**. But how can that be? The universes are not in any receptacle. They have no arrangement.

Alex:
I’m getting a bit confused as to exactly what M* and M** are. Would the following work as an example that you think is the same but I think is different: M* = v, u1, u2, u3… vs M** = v, u2, u3… (where u1 = u2 = u3…)?
In general though, I don’t think that an arrangement is needed to know that adding 1 results in more worlds; subtracting 1 results in fewer worlds. (That said, comparing multiverses where there is no relative conception of adding and subtracting to link them, well, that may be no good. Perhaps that is part of the disagreement?)